MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  tgbtwntriv2 Structured version   Visualization version   GIF version

Theorem tgbtwntriv2 26436
Description: Betweenness always holds for the second endpoint. Theorem 3.1 of [Schwabhauser] p. 30. (Contributed by Thierry Arnoux, 15-Mar-2019.)
Hypotheses
Ref Expression
tkgeom.p 𝑃 = (Base‘𝐺)
tkgeom.d = (dist‘𝐺)
tkgeom.i 𝐼 = (Itv‘𝐺)
tkgeom.g (𝜑𝐺 ∈ TarskiG)
tgbtwntriv2.1 (𝜑𝐴𝑃)
tgbtwntriv2.2 (𝜑𝐵𝑃)
Assertion
Ref Expression
tgbtwntriv2 (𝜑𝐵 ∈ (𝐴𝐼𝐵))

Proof of Theorem tgbtwntriv2
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 simprl 771 . . 3 (((𝜑𝑥𝑃) ∧ (𝐵 ∈ (𝐴𝐼𝑥) ∧ (𝐵 𝑥) = (𝐵 𝐵))) → 𝐵 ∈ (𝐴𝐼𝑥))
2 tkgeom.p . . . . . 6 𝑃 = (Base‘𝐺)
3 tkgeom.d . . . . . 6 = (dist‘𝐺)
4 tkgeom.i . . . . . 6 𝐼 = (Itv‘𝐺)
5 tkgeom.g . . . . . . 7 (𝜑𝐺 ∈ TarskiG)
65ad2antrr 726 . . . . . 6 (((𝜑𝑥𝑃) ∧ (𝐵 𝑥) = (𝐵 𝐵)) → 𝐺 ∈ TarskiG)
7 tgbtwntriv2.2 . . . . . . 7 (𝜑𝐵𝑃)
87ad2antrr 726 . . . . . 6 (((𝜑𝑥𝑃) ∧ (𝐵 𝑥) = (𝐵 𝐵)) → 𝐵𝑃)
9 simplr 769 . . . . . 6 (((𝜑𝑥𝑃) ∧ (𝐵 𝑥) = (𝐵 𝐵)) → 𝑥𝑃)
10 simpr 488 . . . . . 6 (((𝜑𝑥𝑃) ∧ (𝐵 𝑥) = (𝐵 𝐵)) → (𝐵 𝑥) = (𝐵 𝐵))
112, 3, 4, 6, 8, 9, 8, 10axtgcgrid 26412 . . . . 5 (((𝜑𝑥𝑃) ∧ (𝐵 𝑥) = (𝐵 𝐵)) → 𝐵 = 𝑥)
1211adantrl 716 . . . 4 (((𝜑𝑥𝑃) ∧ (𝐵 ∈ (𝐴𝐼𝑥) ∧ (𝐵 𝑥) = (𝐵 𝐵))) → 𝐵 = 𝑥)
1312oveq2d 7189 . . 3 (((𝜑𝑥𝑃) ∧ (𝐵 ∈ (𝐴𝐼𝑥) ∧ (𝐵 𝑥) = (𝐵 𝐵))) → (𝐴𝐼𝐵) = (𝐴𝐼𝑥))
141, 13eleqtrrd 2837 . 2 (((𝜑𝑥𝑃) ∧ (𝐵 ∈ (𝐴𝐼𝑥) ∧ (𝐵 𝑥) = (𝐵 𝐵))) → 𝐵 ∈ (𝐴𝐼𝐵))
15 tgbtwntriv2.1 . . 3 (𝜑𝐴𝑃)
162, 3, 4, 5, 15, 7, 7, 7axtgsegcon 26413 . 2 (𝜑 → ∃𝑥𝑃 (𝐵 ∈ (𝐴𝐼𝑥) ∧ (𝐵 𝑥) = (𝐵 𝐵)))
1714, 16r19.29a 3200 1 (𝜑𝐵 ∈ (𝐴𝐼𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1542  wcel 2114  cfv 6340  (class class class)co 7173  Basecbs 16589  distcds 16680  TarskiGcstrkg 26379  Itvcitv 26385
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2711  ax-nul 5175
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2541  df-eu 2571  df-clab 2718  df-cleq 2731  df-clel 2812  df-ral 3059  df-rex 3060  df-rab 3063  df-v 3401  df-sbc 3682  df-dif 3847  df-un 3849  df-in 3851  df-ss 3861  df-nul 4213  df-sn 4518  df-pr 4520  df-op 4524  df-uni 4798  df-br 5032  df-iota 6298  df-fv 6348  df-ov 7176  df-trkgc 26397  df-trkgcb 26399  df-trkg 26402
This theorem is referenced by:  tgbtwncom  26437  tgbtwntriv1  26440  tgcolg  26503  legid  26536  hlid  26558  lnhl  26564  tglinerflx2  26583  mirreu3  26603  mirconn  26627  symquadlem  26638  outpasch  26704  hlpasch  26705
  Copyright terms: Public domain W3C validator